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A068069
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a(n) = least k which is the start of n consecutive integers each with a different number, 1 through n, of prime factors.
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3
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OFFSET
| 0,2
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COMMENTS
| a(n) >= n!. If the canonical factorization of k is the product of p^e(p) over primes, then the number of distinct number of prime factors is simply the count of the number of p's.
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REFERENCES
| J.-M. De Koninck, J. B. Friedlander, and F. Luca, On strings of consecutive integers with a distinct number of prime factors, Proc. Amer. Math. Soc., 137 (2009), 1585-1592.
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EXAMPLE
| a(0) = 1 because 1 has no prime factors; a(1) = 2 because 2 has the single prime factor 2; a(2) = 5 because 5 = 5^1 & 6 = 2*3 which have 1 & 2 prime factors respectively; a(3) = 28 because 28 = 2^2*7^1, 29 = 29^1 & 30 = 2*3*5 which have 2, 1 & 3 prime factors respectively; a(4) = 417 because 417 = 3*139, 418 = 2*11*19, 419 = 419^1 & 420 = 2^2*3*5*7 which have 2, 3, 1 & 4 prime factors (distinct) respectively and this represents a record-breaking number.
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MATHEMATICA
| k = 3; Do[k = k - n; a = Table[ Length[ FactorInteger[i]], {i, k, k + n - 1}]; b = Table[i, {i, 1, n}]; While[ Sort[a] != b, k++; a = Drop[a, 1]; a = Append[a, Length[ FactorInteger[k]]]]; Print[k - n + 1], {n, 1, 7}]
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CROSSREFS
| Cf. A067665.
Sequence in context: A019043 A009635 A138293 * A105787 A110497 A000472
Adjacent sequences: A068066 A068067 A068068 * A068070 A068071 A068072
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KEYWORD
| more,nonn
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 20 2002
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EXTENSIONS
| One more term from Labos E. (labos(AT)ana.sote.hu), May 26 2003
One more term from Donovan Johnson (donovan.johnson(AT)yahoo.com), Apr 03 2008
Corrected example and a(9) from Donovan Johnson (donovan.johnson(AT)yahoo.com), Aug 31 2010
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